3 Answers
3

These are are completely different concepts, which sometimes may be connected.

A motif in biology is a mathematical model, typically of a sequence, which is predictive of which sequences to some defined group. For example, a DNA sequence motif can characterize the binding site of a transcription factor, i.e. which sequences tend to be bound by this factor. For proteins, sequence motifs can characterize which proteins (protein sequences) belong to a given protein family. A simple motif could be, for example, some pattern which is strictly shared by all members of the group, e.g. WTRXEKXXY (where X stands for any amino acid). There are also more complex motif models.

Protein domains, on the other hand, are a structural entity, usually meaning a part of the protein structure which folds and functions independently. So, proteins are often constructed from different combinations of these domains.

So how are motifs and domains related? Well, when you think about protein families, it makes sense not only to look at the whole sequence but also to focus on individual domains. Since they are a elementary functional-structural units, it makes sense to find sequence motifs for individual domains. So, you often find that a protein contains multiple domains, each domain characterized by having a sequence that matches the motif of its family.

adding to the answer of @Bitwise: Motif is just a statistically well represented pattern in a data. It can be a sequence or even a structure. Motifs CAN be sometimes associated with particular functions
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WYSIWYGApr 8 '13 at 12:56

@WYSIWYG thanks for the clarification, I was going to mention structural motifs but forgot to write it.
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BitwiseApr 8 '13 at 18:08

This was very helpful, thanks for the reply. :)
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LaurenApr 9 '13 at 19:31

The second, equally common, use of the term motif refers to a set of
contiguous secondary structure elements that either have a particular
functional significance or define a portion of an independently folded
domain.

After the peptide chain of a protein has been organised into successive stretches of secondary elements, combinations of such elements are first arranged into distinctive groups called super secondary motifs. Several neighbouring secondary elements and supersecondary motifs are folded and twisted further into compact globular cluster called a domain.